Derivative of sqrt(sin(5*x))

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  __________
\/ sin(5*x)

$$\sqrt{\sin{\left(5 x \right)}}$$

sqrt(sin(5*x))

Detail solution

Let $u = \sin{\left(5 x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 x \right)}$ :
1. Let $u = 5 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $5$
  The result of the chain rule is:
  
  $5 \cos{\left(5 x \right)}$
The result of the chain rule is:

$\frac{5 \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$

The answer is:

$\frac{5 \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$

The graph

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The first derivative [src]

  5*cos(5*x)  
--------------
    __________
2*\/ sin(5*x)

$$\frac{5 \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$$

Simplify

The second derivative [src]

    /                     2      \
    |    __________    cos (5*x) |
-25*|2*\/ sin(5*x)  + -----------|
    |                    3/2     |
    \                 sin   (5*x)/
----------------------------------
                4

$$- \frac{25 \left(2 \sqrt{\sin{\left(5 x \right)}} + \frac{\cos^{2}{\left(5 x \right)}}{\sin^{\frac{3}{2}}{\left(5 x \right)}}\right)}{4}$$

Simplify

The third derivative [src]

    /         2     \         
    |    3*cos (5*x)|         
125*|2 + -----------|*cos(5*x)
    |        2      |         
    \     sin (5*x) /         
------------------------------
            __________        
        8*\/ sin(5*x)

$$\frac{125 \left(2 + \frac{3 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \cos{\left(5 x \right)}}{8 \sqrt{\sin{\left(5 x \right)}}}$$

Simplify

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Derivative of sqrt(sin(5*x))

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The solution